From Wikipedia, the free encyclopedia

The two person bargaining problem is a problem of understanding
how two agents should cooperate when non-cooperation leads to Pareto-inefficient results. It is in
essence an equilibrium selection problem; Many games have multiple
equilibria with varying payoffs for each player, forcing the
players to negotiate on which equilibrium to target. The
quintessential example of such a game is the Ultimatum game.
The underlying assumption of bargaining theory is that the
resulting solution should be the same solution an impartial
arbitrator would recommend. Solutions to bargaining come in two
flavors: an axiomatic approach where desired properties of a
solution are satisfied and a strategic approach where the
bargaining procedure is modeled in detail as a sequential game.

An
example

Opera

Football

Opera

3,2

0,0

Football

0,0

2,3

Battle of the Sexes
1

The Battle of the Sexes, as shown, is a two
player coordination game. Both Opera/Opera
and Football/Football are Nash equilibria.
Any probability distribution over these two Nash equilibria is a correlated equilibrium. The
question then becomes which of the infinite possible equilibria
should be chosen by the two players. If they disagree and choose
different distributions then they will fail to coordinate and
likely receive 0 payoffs. In this symmetric case the natural choice
is to play Opera/Opera and Football/Football with even probability.
Indeed all bargaining solutions described below prescribe this
solution. However if the game is asymmetric (for example
Football/Football instead yields payoffs of 2,5) the appropriate
distribution becomes less clear. Bargaining theory solves this
problem.

The
Formal Description

A 2 person bargain problem consists of a disagreement point
v (also known as a threat
point) and a feasibility set F. v
= (v1,v2), where
v1 and v2 are the payoffs after
disagreement to player 1 and player 2 respectively. F is a closed convex subset of
representing the set of possible agreements. F is convex because an agreement
could take the form of a correlated combination of other
agreements. Points in F must
all be better than the disagreement point as there is no sense to
an agreement which is worse than disagreement. The goal of
bargaining is to choose the feasible agreement φ in F
that would result after thorough negotiations.

Advertisements

Feasibility
Set

The set of possible agreements F depends on if there is an outside
regulator affording binding contracts. When binding contracts are
allowed any joint action is playable so the feasibility set
consists of all attainable payoffs better than the disagreement
point. When binding contracts are not allowed the game is said to
have moral hazard (as players can defect) and thus the feasibility
set only consists of correlated equilibrium, which need no
enforcement.

Disagreement
Point

The disagreement point v
is the value the players can expect to receive if negotiations
break down and no bargain can be reached. Naively this could just
be some focal equilibrium which both
players could expect to play. However, this point directly affects
eventual bargaining solution, so it stands to reason that each
player should attempt to choose their disagreement points in order
to maximize their bargaining position. Towards this goal, it is
often advantageous to simultaneously increase one’s own
disagreement payoff while harming one’s opponent's disagreement
payoff - hence this point is often known as the threat point. If
threats are viewed as actions then we can construct a separate game
where each player chooses a threat and receives a payoff according
to the outcome of bargaining. This is known as Nash’s variable
threat game. Alternatively each player could play a minimax strategy in case of
disagreement, choosing to disregard personal reward in order to
hurt the opponent as much as possible if they leave the bargaining
table.

Bargaining
Solutions

Various solutions have been proposed based on slightly different
assumptions about what properties are desired for the final
agreement point.

Nash
bargaining solution

John Nash proposed that a solution should satisfy certain
axioms, 1) Invariant to affine transformations or Invariant to
equivalent utility representations, 2) Pareto
optimality, 3) Independence of
irrelevant alternatives, 4) Symmetry. Let us call u the utility
function for player 1, v the utility function for player
2. Under these conditions, rational agents will choose what is
known as the Nash bargaining solution. Namely, they will
seek to maximize | u(x) −
u(d) | | v(y) −
v(d) | , where u(d) and v(d), are the status quo utilities
(i.e. the utility obtained if one decides not to bargain with the
other player). The product of the two excess utilities is generally
referred to as the Nash product.

Kalai-Smorodinsky
bargaining solution

Independence of Irrelevant Alternatives can be substituted with
an appropriate monotonicity condition, thus providing a different
solution for the class of bargaining problems. This alternative
solution has been introduced by Ehud Kalai and Meir Smorodinsky. It
is the point which maintains the ratios of maximal gains. In other
words, if player 1 could receive a maximum of g1 with player 2’s help
(and vice-versa for g2), then the
Kalai-Smorodinsky bargaining solution would yield the point φ on the Pareto frontier such that φ1 / φ2 =
g1 / g2 .

Egalitarian bargaining
solution

The egalitarian bargaining solution, introduced by Ehud Kalai,
is a third solution which drops the condition of scale invariance
while including both the axiom of Independence of
irrelevant alternatives, and the axiom of monotonicity. It is
the solution which attempt to grant equal gain to both parties.

Applications

Recently the Nash bargaining game has been used by some philosophers and economists in order to
explain the emergence of human attitudes toward distributive justice (Alexander
2000; Alexander and Skyrms 1999; Binmore 1998, 2005). These authors
primarily use evolutionary game theory in
order to explain how individuals come to believe that proposing a
50-50 split is the only just
solution to the Nash Bargaining Game.